Involution surface bundles over surfaces
- PDF / 364,095 Bytes
- 20 Pages / 439.37 x 666.142 pts Page_size
- 29 Downloads / 214 Views
Mathematische Zeitschrift
Involution surface bundles over surfaces Andrew Kresch1 · Yuri Tschinkel2,3 Received: 4 July 2018 / Accepted: 10 December 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We construct models of involution surface bundles over algebraic surfaces, degenerating over normal crossing divisors, and with controlled singularities of the total space.
1 Introduction In this paper, we continue our investigation of del Pezzo fibrations, initiated in [7]. Here we treat the case of fibrations in minimal del Pezzo surfaces of degree 8, also known as involution surfaces. Over the generic point of the base they are forms of quadric surfaces; see [2] for an arithmetic description. Quadric surface bundles constitute a special case. The main result of [1] gives a dictionary between quadric surface bundles X → S, where S is regular and has dimension at most 2, degenerating along a regular divisor to A1 -singular quadrics, and sheaves of Azumaya quaternion algebras on a double cover of S. The approach taken there is based on Clifford algebras and orders in quaternion algebras. Our results are more general in several respects: • We allow more general degenerations, including quadric surface fibrations where the defining equation drops rank by 2 along a regular divisor (Definition 1). • We allow the base S to have arbitrary dimension (Theorem 6). • When S is a surface over an algebraically closed field, we allow degeneration along an arbitrary divisor and construct (Theorem 2) birational models with controlled singularities degenerating over normal crossing divisors. Definition 1 Let S be a regular scheme, in which 2 is invertible in the local rings. An involution surface bundle over S is a flat projective morphism π : X → S such that the locus U ⊂ S over which π is smooth is dense in S and the fibers of π over points of U are involution surfaces. An involution surface bundle π : X → S is said to have mild degeneration if every singular fiber is geometrically isomorphic to one of the following reduced schemes:
B
Yuri Tschinkel [email protected] Andrew Kresch [email protected]
1
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland
2
Courant Institute, 251 Mercer Street, New York, NY 10012, USA
3
Simons Foundation, 160 Fifth Avenue, New York, NY 10010, USA
123
A. Kresch, Y. Tschinkel
Type I: a quadric surface with an A1 -singularity; Type II: the self-product of a reduced singular conic; Type III: a union of two copies of the Hirzebruch surface F2 , each with (−2)-curve glued to a fiber of the other; Type IV: the product with P1 of a reduced singular conic. Theorem 2 Let k be an algebraically closed field of characteristic different from 2, S a smooth projective surface over k, and π: X → S a morphism of projective varieties whose generic fiber is an involution surface. Then there exists a commutative diagram X
X
X π
π˜
S
S
S
such that S, • S is a birational morphism from a smooth projective surface •
Data Loading...
